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PHYSICAL REVIEW D, VOLUME 59, 074020Vector meson dominance and ther meson
M. Benayoun*LPNHE des Universites Paris VI et VIIIN2P3, Paris, France
H. B. OConnell
Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 405060055
A. G. Williams
Department of Physics and Mathematical Physics and Special Research Centre for the Subatomic Structure of Matter,University of Adelaide 5005, Australia
~Received 29 July 1998; published 8 March 1999!
We discuss the properties of vector mesons, in particular ther0, in the context of the hidden local symmetry~HLS! model. This provides a unified framework to study several aspects of the low energy QCD sector. First,we show that in the HLS model the physical photon is massless, without requiring off field diagonalization. Wethen demonstrate the equivalence of HLS and the two existing representations of vector meson dominance,VMD1 and VMD2, at both the tree level and one loop order. Finally theS matrix pole position is shown toprovide a model and process independent means of specifying ther mass and width, in contrast with the realaxis prescription currently used in the Particle Data Group tables.@S05562821~99!028076#
PACS number~s!: 12.40.Vv, 11.30.Qc, 11.30.Rd, 12.39.Fewth
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I. INTRODUCTION
There is no reliable analytic means for calculating loand medium energy strongly interacting processes withunderlying theory, QCD. Despite progress in numerical sties of QCD both via the lattice@1# and QCDmotivated models @2#, preQCD effective Lagrangians involving the oserved hadronic spectrum continue to play an importantin studies of this sector. We shall be concerned in this wwith the interactions of the pseudoscalar and vector mesas described by the hidden local symmetry~HLS! model@3#.
The focus of our paper will be ther resonance. As thelightest and broadest of the vector octets it plays an imptant role in phenomenology and is presently the subjecinterest as a possible indicator of chiral symmetry restorain heavy ion collisions@4#. It also serves as a guide for phyics in other sectors. As we shall see the interaction of vemesons and photons in the HLS construction is closanalogous to the SU~2!^U~1! symmetry breaking of theelectroweak interaction. The traditional determination ofr mass and width has been plagued by model dependewhich as we show can be avoided by use of theSmatrixpole, closely following developments in the study of theZ0.
Our paper is structured as follows: we begin with a broutline of the HLS model and discuss the generation of vtor boson masses by the spontaneous breaking of the gchiral symmetry through the HiggsKibble~HK! mechanism@5#. In Sec. III we consider the relationship between theHK masses and thephysical vector masses, using a twochannel propagator matrix for the photonr system. This allows one to consider the effect of mixing in the dressingthe bare propagators and when this is properly considered
*Email address: benayoun@in2p3.frEmail address: hoc@pa.uky.eduEmail address: awilliam@physics.adelaide.edu.au05562821/99/59~7!/074020~11!/$15.00 59 0740e
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dressed photon is seen to be massless as required, withe need for a change of basis.
Section IV is devoted to a comparison of the two comonly used representations of vector meson domina~VMD ! ~referred to hereafter as VMD1 and VMD2 following the convention of Ref.@6#!, both of which can be obtained as special cases from the HLS Lagrangian of Ref.@3#.Note that VMD2 is the most commonly used version andoften in the literature, simply referred to as VMD or thvector dominance model@7#. Using the pion and kaon formfactors we explicitly demonstrate their equivalence at the tlevel ~which is trivial! and at oneloop order, where carerequired. This treatment is performed in the general casa2, wherea is the HLS parameter@3#, for which we introduce VMD1a and VMD2a in an obvious manner.
Finally, as the vector mesons are resonant particles,study the effect of the large width on the determinationmodel independentr parameters. TheSmatrix pole positionis shown to provide a truly modelindependent and, furthmore, processindependent parametrization of ther meson.
II. HIDDEN LOCAL SYMMETRY AND VMD
The HLS model allows us to produce a theory with vecmesons as the gauge bosons of a hidden local symmThese then become massive because of the spontanbreaking of a chiralU(3)L ^ U(3)R global symmetry. Let usconsider the chiral Lagrangian@8#
Lchiral51
4Tr @]mF]
mF#, ~1!
whereF(x)5 f pU(x) in the usual notation. This exhibits thchiral U(3)L ^ U(3)R symmetry underUgLUgR . We canwrite this in exponential form and expand
F~x!5 f pe2iP~x!/ f p5 f p12iP~x!22P
2~x!/ f p1 ;~2!1999 The American Physical Society201
t aing,n by thes
M. BENAYOUN, H. B. OCONNELL, AND A. G. WILLIAMS PHYSICAL REVIEW D 59 074020therefore, substituting into Eq.~1! we see that the vacuum corresponds toP50, U51. That is,F has a nonzero vacuumexpectation value which spontaneously breaks theU(3)L ^ U(3)R symmetry as this symmetry of the Lagrangian is nosymmetry of the vacuum. The massless Goldstone bosons contained inP, associated with the spontaneous symmetry breakthen correspond to the perturbations about the QCD vacuum and we can think of expansions in this field as giveHermitian matrixP5PaTa where the GellMann matrices are normalized such that Tr@TaTb#5dab/2. So for the pseudoscalarone has
P51
& S 1& p01 1A6 p81 1) h0 p1 K1p2 2 1& p01 1A6 p81 1) h0 K0K2 K0 2A2
3p81
1
)h0
D . ~3!rge
t
tisde

try,he
a
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eenHowever, in addition to the global chiral symmetry,G,the HLS scheme includes a local symmetry,H, in Eq. ~1! inthe following way@3#. Let
U~x![jL~x!jR~x! ~4!
where
jR,L~x!5exp@ iS~x!/ f S#exp@6 iP~x!/ f p# ~5!
jR,L~x!h~x!jR,L~x!gR,L . ~6!
Note that this introduces the scalar field,S(x), analogous tothe pseudoscalar,P(x), of Eq. ~3!, though S(x) does notappear in the chiral fieldU(x) of Eq. ~4!. The general formsof the transformations are given bygL,R5exp(ieL,R
a Ta) andh(x)5exp@ieH
a (x)Ta#.One now seeks to incorporate HLS into the low ene
Lagrangian in a nontrivial way, thereby introducing thlightest vector meson states@3,9#. The procedure is to firsrewriteLchiral explicitly in terms of thej components:
Lchiral52f p
2
4Tr@]mjLjL
2]mjRjR #2. ~7!
The Lagrangian can be gauged for both electromagneand the hidden local symmetry by changing to covariantrivatives
DmjL,R5]mjL,R2 igVmjL,R1 iejL,RAmQ ~8!
whereAm is the photon field,Q5diag(2/3,21/3,21/3) andVm5Vm
a Ta whereVma are the vector meson fields transform
ing as Vmh(x)Vmh(x)1 ih(x)]mh(x)/g. Suppressingfor brevity the spacetime indexm, we can write the vectormeson field matrixVm as
V51
& S ~r01v!/& r1 K* 1r2 ~2r01v!/& K* 0K* 2 K* 0 f
D . ~9!
07402y
m
Here we have used ideal mixing in defining the barev andfmesons. Note that the vector meson fieldsVm
a 5K* ,r,v,fare introduced in the role of gauge fields forH[flavorSU~3!. However,LA[Lchiral is independent ofV, and in theHLS model a second piece of the Lagrangian,LV , is introduced:
LA52f p
2
4Tr@~DmjLjL
2DmjRjR !#2
LV52f p
2
4Tr@~DmjLjL
1DmjRjR !#2. ~10!
The full HLS Lagrangian is then finally defined by@3#
LHLS5LA1aLV , ~11!
where we see that the HLS parametera has now been introduced.
In the absence of the gauge fields,Vm andAm , we see thatEq. ~10! reduces to
LA51
2Tr@]mP]
mP#, LV5f p
2
2 f S2 Tr@]mS]
mS# ~12!
to quadratic order in bosons. In this caseP and S, wouldboth be Goldstone bosons, whereP(x) is associated with thespontaneous breaking of the usual, global chiral symmeG, of LA arising from the vacuum expectation value of tU(x) fields and analogously forS(x). However, gauge invariance allows for their elimination. It is usual to takespecial gauge, the unitary gauge, forH, for which the scalarfield no longer appears,S(x)50 @9# ~for a discussion of theunitary gauge and spontaneously broken symmetries, seexample Sec. 1253 of Ref.@10#!. This is phenomenologically reasonable as no chiral partner for the pion has bobserved. With this choice we have
jL~x!5jR~x![j~x!5exp@ iP~x!/ f p#. ~13!02
g
nl
ra
arof
s
VECTOR MESON DOMINANCE AND THEr MESON PHYSICAL REVIEW D59 074020By demandingS(x)50 the local symmetry,H, is lost, buttheg transformations of the global chiral symmetry groupGwill regenerate the scalar field through
j~x!j8~x!5j~x!g
5exp@ iS8P~x!,g/ f S#exp@ iP8~x!/ f p#. ~14!
However, the system can still maintain the unitarity gauconditionS(x)50 and global chiral symmetry,G, under thecombined transformation@9#
j~x!j8~x!5hP~x!,gj~x!g,
hP~x!,g5exp@2 iS8P~x!,g/ f S# ~15!
where the particular choice of local transformatiohP(x),g, kills the scalar field created by the globa07402e
,
transformationg. The physical meaning of this is that vectofields acquire longitudinal components by eating the sclar S field through a transformation of the form~to lowestorder in Goldstone fields!
VmVm21
g fS]mS8. ~16!
Once the scalar field is effectively removed by the unitity gauge choice the traditional VMD Lagrangian, thatVMD2, is obtained from an expansion ofLHLS in the pionfield, as per Eq.~2!. The HLS model actually generalizeVMD2, through the add